The coordinate system

Equatorial coordinates are fixed on the celestial sphere, following the daily
rotation of the Earth and its yearly revolution about the Sun. Because of
this, the main requirement for locating an object upon the sky is simply
the local mean sidereal time, and the direction of the north celestial pole
(approximately coincident with Polaris). I said main
requirement because one also must know the epoch of the coordinate
system. Because of the Earth's precession, the right ascension and
declination change by several arcseconds per year. Common epochs
you'll find in modern catalogs are B1950 and J2000, which give the equatorial
coordinate positions measured in the years 1950 and 2000. (The "B" and "J"
stand for Besselian and Julian.) One should also
realize that the "celestial sphere" is not truly a fixed sphere of stars, and
that stars themselves have proper motion through space. Some stars have
very large proper motions -- nearby stars can move several arcseconds per
year -- so one may also need to account for this.

Getting there

To determine the apparent position of an object on the sky given its
equatorial coordinates, first determine the local mean sidereal time, which
is the right ascension of the meridian -- the north-south arc in the sky
that runs directly overhead. From this, you can obtain the hour angle, which
is the number of hours east or west of the meridian the object is. The hour
angle, HA, is defined by

where LMST is the local mean sidereal time.
As an example, suppose you have a sidereal clock which keeps track of your
local mean sidereal time, and it says the time is 02:00:00. The object you're
interested in has a right ascension of 04:30:00. Therefore, the hour angle
is -02:30:00, or two and a half hours east of the meridian. (This means
the star will cross your local meridian in about two and a half hours.)

The declination is simply the distance in degrees due north or south of the
celestial equator. Again, assume the object has a declination of +16° 00'
00".
If you draw an hour circle -- a circle at the hour angle passing through
the north and south celestial poles --
you will find the object 16 degrees north of the intersection of this hour
circle and the celestial equator.

The practical (if tedious) method for locating the apparent point on the sky of a set of equatorial coordinates is to use set of trigonometric equations which
convert equatorial coordinates to altazimuth coordinates.
You need to know the hour angle, your latitude, and the equatorial coordinates
of the object. These equations are:

In practice, the chances are that you will have a star chart that will easily help you find your way to the object you're interested
in based on some easily spotted reference stars, or else you're lucky/well-to-do
enough to have a telescope with an equatorial mount, which makes computing
the altaz coordinates unnecessary. However, the equations given above are
used frequently in research astronomy for (at least) two reasons. First,
they're a convenient means of figuring out when your object of research
interest will be visible in the sky. Second, most modern, large telescopes
do not use equatorial mounts, but instead use altazimuth mounts to save on
cost and size. The telescope then tracks positions on the sky based on these
equations, rather than simply fixing the declination and allowing
the telescope to follow the right ascension.

Origins of the system

The time-frame of the invention of equatorial coordinates is apparently in
some dispute. Some suggest that equatorial coordinates were known and used
as early as the time of Hipparchus. In his Commentary on Aratus (In Arati
et Eudoxi phaenomena commentariorium), Hipparchus apparently used what
appear to be the right ascension and declination of several bright stars
whose positions are correct if the modern coordinates are precessed back to
150 BC. Regardless of whether this is the case, equatorial coordinates
clearly go back before the origins of the telescope around
the turn of the 17th century. The system was known and used by Arabic
astronomers and was certainly well-known and used by the time of
Jamshid al-Kashi and Ulugh Beg. Equatorial mounts and alignments were
used even for pre-telescopic astronomical instruments, and
equatorially-mounted telescopes (such as the English or Yoke mount) were
likely invented before the mid 18th century.